Algebraic and analytic geometry of fans.

Abstract

A set which can be defined by systems of polynomial inequalities is called semialgebraic. When such a scription is possible locally around every point, by means of analytic inequalities varying with the point, the set; is called semianalytic. If one single system of strict inequalitites is enough, either globally or locally at every point, the set is called basic. The topic of this work is the relationship between these two notions. Namely, we describe and characterize, both algebraically and geometrically, the obstructions for a basic semianalytic set to be basic semialgebraic. Then, we describe a special family of obstructions that suffices to recognize whether or not a basic semianalytic set is basic semialgebraic. Finally, we use the preceding results to discuss the effect on basicness of birational transformations.